L(s) = 1 | + (−1.30 + 0.136i)2-s + (2.02 + 1.82i)3-s + (−0.277 + 0.0590i)4-s + (0.166 + 0.373i)5-s + (−2.89 − 2.10i)6-s + (−2.50 + 0.847i)7-s + (2.84 − 0.924i)8-s + (0.466 + 4.43i)9-s + (−0.268 − 0.464i)10-s + (3.26 − 0.590i)11-s + (−0.671 − 0.387i)12-s + (0.864 − 0.628i)13-s + (3.14 − 1.44i)14-s + (−0.345 + 1.06i)15-s + (−3.06 + 1.36i)16-s + (0.576 − 5.48i)17-s + ⋯ |
L(s) = 1 | + (−0.921 + 0.0968i)2-s + (1.17 + 1.05i)3-s + (−0.138 + 0.0295i)4-s + (0.0744 + 0.167i)5-s + (−1.18 − 0.858i)6-s + (−0.947 + 0.320i)7-s + (1.00 − 0.326i)8-s + (0.155 + 1.47i)9-s + (−0.0847 − 0.146i)10-s + (0.984 − 0.177i)11-s + (−0.193 − 0.111i)12-s + (0.239 − 0.174i)13-s + (0.841 − 0.386i)14-s + (−0.0891 + 0.274i)15-s + (−0.765 + 0.340i)16-s + (0.139 − 1.33i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.634300 + 0.425042i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.634300 + 0.425042i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (2.50 - 0.847i)T \) |
| 11 | \( 1 + (-3.26 + 0.590i)T \) |
good | 2 | \( 1 + (1.30 - 0.136i)T + (1.95 - 0.415i)T^{2} \) |
| 3 | \( 1 + (-2.02 - 1.82i)T + (0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 + (-0.166 - 0.373i)T + (-3.34 + 3.71i)T^{2} \) |
| 13 | \( 1 + (-0.864 + 0.628i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.576 + 5.48i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (4.47 + 0.951i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-1.37 + 2.38i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.06 - 0.345i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (2.75 - 6.19i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (4.49 + 4.98i)T + (-3.86 + 36.7i)T^{2} \) |
| 41 | \( 1 + (-2.68 - 8.25i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 4.85iT - 43T^{2} \) |
| 47 | \( 1 + (0.543 - 2.55i)T + (-42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (12.2 + 5.46i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (0.435 + 2.04i)T + (-53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (6.52 - 2.90i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-4.12 - 7.14i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.08 - 1.51i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (9.49 - 2.01i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-2.06 + 0.216i)T + (77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-2.94 - 2.14i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-2.68 - 1.54i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.81 + 10.7i)T + (-29.9 + 92.2i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.68232897359385981465831267261, −13.93781831284292666214029413902, −12.70712836849013812014167208508, −10.82127078937091647455426082120, −9.765871695319519583425907430896, −9.130955495223839229253944009262, −8.406965209599221549111441260255, −6.78382026346244788826637026092, −4.49305777751897913524761340342, −3.08534918918524077249302156937,
1.64216262196223965143224546463, 3.78085505659539078681908827795, 6.47383666569848775898362221573, 7.60942327288435874159300298346, 8.733012438515680511818774998878, 9.340419378016792309164689378775, 10.63494010911726838943619735086, 12.47124203179873414525848772942, 13.24731200734568900036055375804, 14.08295078735653357299130488768